Subgroups — #65686

Order 320.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×AGL(1,𝔽₅)	C₂	1	2	yes	maximal
2	C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	(C₂×AGL(1,𝔽₅))⋊C₄	C₂	1	4	yes	maximal
4	C₂²×AGL(1,𝔽₅)	C₂²	1	4	yes	—
5	C₂×C₄×D₅	C₂²	1	2	yes	—
6	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₄×AGL(1,𝔽₅)	C₄	1	8	yes	complemented
8	C₄²×C₅	C₄	1	1	yes	characteristic, complemented
9	C₄×Dic₅	C₄	1	2	yes	complemented
10	C₄×Dic₅	C₄	1	1	yes	characteristic, complemented
11	#376	—	5	1	no	maximal, Sylow
12	C₂²×D₅	C₂³	1	1	yes	characteristic
13	C₂×AGL(1,𝔽₅)	—	2	8	no	—
14	C₂×AGL(1,𝔽₅)	C₂×C₄	1	8	yes	—
15	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₅	C₂×C₄	1	2	yes	—
18	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₄×D₅	—	2	4	no	—
20	C₄×D₅	C₂×C₄	1	4	yes	—
21	C₂×C₄²	—	5	2	no	—
22	C₂×C₄²	—	5	1	no	—
23	#95	—	5	4	no	—
24	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
25	C₂×D₅	C₂²×C₄	1	2	yes	—
26	C₂×D₅	#41	1	4	yes	—
27	AGL(1,𝔽₅)	—	2	8	no	—
28	C₄×C₅	—	2	2	no	—
29	C₄×C₅	C₄²	1	2	yes	—
30	Dic₅	—	2	2	no	—
31	Dic₅	C₄²	1	2	yes	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	4	no	—
34	C₂²×C₄	—	5	2	no	—
35	C₄²	—	5	8	no	—
36	C₄²	—	5	2	no	—
37	C₄²	—	5	1	no	—
38	C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
39	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
40	C₂×C₅	#117	1	2	yes	—
41	D₅	#117	1	4	yes	—
42	C₂³	—	5	1	no	—
43	C₂×C₄	—	5	1	no	—
44	C₂×C₄	—	5	4	no	—
45	C₂×C₄	—	5	8	no	—
46	C₂×C₄	—	5	2	no	—
47	C₂×C₄	—	10	4	no	—
48	C₂×C₄	—	10	8	no	—
49	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
50	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₅	#376	1	1	yes	characteristic, complemented, Sylow
52	C₂²	—	5	2	no	—
53	C₂²	—	5	4	no	—
54	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	5	2	no	—
57	C₄	—	10	8	no	—
58	C₄	—	10	2	no	—
59	C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
60	C₂	—	5	4	no	—
61	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂	#4130	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical, cosocle	C₂²×D₅	C₂³
hypercenter	C₄²	AGL(1,𝔽₅)
hypocenter	C₅	#376
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₄×AGL(1,𝔽₅)	C₄	semidirect
8. C₄²×C₅	C₄	semidirect
9. C₄×Dic₅	C₄	semidirect
10. C₄×Dic₅	C₄	semidirect
38. C₄²	AGL(1,𝔽₅)	semidirect
51. C₅	#376	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×AGL(1,𝔽₅)	2
C₄²×D₅	2
(C₂×AGL(1,𝔽₅))⋊C₄	2
#376	5